Abstract

In this note, we provide upper bounds on the expectation of the supremum of empirical processes indexed by Hölder classes of any smoothness and for any distribution supported on a bounded set in \(\mathbb{R}^{d}\). These results can alternatively be seen as non-asymptotic risk bounds, when the unknown distribution is estimated by its empirical counterpart, based on \(n\) independent observations, and the error of estimation is quantified by integral probability metrics (IPM). In particular, IPM indexed by Hölder classes are considered and the corresponding rates are derived. These results interpolate between two well-known extreme cases: the rate \(n^{-1/d}\) corresponding to the Wassertein-1 distance (the least smooth case) and the fast rate \(n^{-1/2}\) corresponding to very smooth functions (for instance, functions from a RKHS defined by a bounded kernel).

中文翻译：

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对由 Hölder 类索引的经验过程的上界的期望有界

摘要

在本说明中，我们提供了由具有任何平滑度的 Hölder 类索引的经验过程的最高期望的上限，以及\(\mathbb{R}^{d}\) 中的有界集支持的任何分布。这些结果也可以被视为非渐近风险界限，当未知分布由其经验对应物估计时，基于\(n\)独立观察，估计误差由积分概率度量 (IPM) 量化。特别是，考虑了由 Hölder 类别索引的 IPM，并推导出相应的比率。这些结果在两个众所周知的极端情况之间进行插值：对应于 Wassertein-1 距离的速率\(n^{-1/d}\)（最不平滑的情况）和快速速率\(n^{-1/2}\)对应于非常平滑的函数（例如，来自有界内核定义的 RKHS 的函数）。